Problem: Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{q^2 - 6q - 16}{q^2 + 5q + 6}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 6q - 16}{q^2 + 5q + 6} = \dfrac{(q - 8)(q + 2)}{(q + 3)(q + 2)} $ Notice that the term $(q + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 2)$ gives: $x = \dfrac{q - 8}{q + 3}$ Since we divided by $(q + 2)$, $q \neq -2$. $x = \dfrac{q - 8}{q + 3}; \space q \neq -2$